Question

An alpha particle and a proton having same momentum enter into a region of uniform magnetic field and move in circular paths . The ratio of the radii of curvature of their path r alpha /r proton in the field is (1) 1/2 (2) 1/4. (3) 1. (4)4

Answer 1

Answer on Question 75044, Physics, Electromagnetism

Question:

An alpha particle and a proton having same momentum enter into a region of uniform magnetic field and move in circular paths. The ratio of the radii of curvature of their path $r_{alpha} / r_{proton}$ in the field is:

a) $1/2$

b) $1/4$

c) 1

d) 4

Solution:

There are two forces that act on the charged particle when it moves in the uniform magnetic field: the magnetic force and the radial force. So, using the Newton's second law of motion we can write:

$$qvB = \frac{mv^2}{r},$$

here, $q$ is the charge of the particle, $v$ is the orbital speed of the particle, $B$ is the magnetic field, $m$ is the mass of the particle, $r$ is the radius of the curvature of particle's path.

From this formula, we can find the radius of the curvature of particle's path:

$$r = \frac{mv}{qB} = \frac{p}{qB},$$

here, $p$ is the momentum of the particle.

Since, the both particles (proton and alpha-particle) have the same momentum and the magnetic field is uniform, we can write:

$$r = \frac{p}{qB} \propto \frac{1}{q}.$$

Finally, we can find $r_{alpha} / r_{proton}$:

$$\frac {r _ {a l p h a}}{r _ {p r o t o n}} = \frac {\frac {1}{q _ {a l p h a}}}{\frac {1}{q _ {p r o t o n}}} = \frac {q _ {p r o t o n}}{q _ {a l p h a}} = \frac {1}{2}.$$

Answer:

a) $\frac{r_{alpha}}{r_{proton}} = \frac{1}{2}$ .

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